Geometric Image Labeling with Global Convex Labeling Constraints

  • Artjom ZernEmail author
  • Karl Rohr
  • Christoph Schnörr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10746)


In [2], a smooth geometric labeling approach was introduced by following the Riemannian gradient flow of a given objective function on the so-called assignment manifold. The approach evaluates a user-defined data term and performs spatial regularization by Riemannian averaging of the assignment vectors. In this paper, we extend this approach in order to impose global convex constraints on the labeling results based on linear filter statistics in the label space. The smoothness of the approach is preserved by using logarithmic barrier functions to handle the new constraints. We discuss how suitable filters can be determined from example data of a given image class, and we demonstrate numerically the effectiveness of the constraints in several academic labeling scenarios.


Image labeling Assignment manifold Statistical label constraints Riemannian gradient flow Information geometry 



We gratefully acknowledge support by the German Science Foundation, grant GRK 1653.


  1. 1.
    Alvarez, F., López, J.: Convergence to the optimal value for barrier methods combined with Hessian Riemannian gradient flows and generalized proximal algorithms. J. Convex Anal. 17(3&4), 701–720 (2010)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Åström, F., Petra, S., Schmitzer, B., Schnörr, C.: Image labeling by assignment. J. Math. Imag. Vis. 58(2), 211–238 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benning, M., Gilboa, G., Grah, J.S., Schönlieb, C.-B.: Learning filter functions in regularisers by minimising quotients. In: Lauze, F., Dong, Y., Dahl, A.B. (eds.) SSVM 2017. LNCS, vol. 10302, pp. 511–523. Springer, Cham (2017). CrossRefGoogle Scholar
  4. 4.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cover, T., Thomas, J.: Elements of Information Theory, 2nd edn. Wiley, Hoboken (2006)zbMATHGoogle Scholar
  6. 6.
    Kappes, J., Andres, B., Hamprecht, F., Schnörr, C., Nowozin, S., Batra, D., Kim, S., Kausler, B., Kröger, T., Lellmann, J., Komodakis, N., Savchynskyy, B., Rother, C.: A comparative study of modern inference techniques for structured discrete energy minimization problems. Int. J. Comp. Vis. 115(2), 155–184 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lowe, D.: Distinctive image features from scale-invariant keypoints. Int. J. Comp. Vis. 60(2), 91–110 (2004)CrossRefGoogle Scholar
  8. 8.
    Morel, J.M., Yu, G.: ASIFT: a new framework for fully affine invariant image comparison. SIAM J. Imag. Sci. 2(2), 438–469 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Nesterov, Y., Nemirovskii, A.: Interior Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)CrossRefzbMATHGoogle Scholar
  10. 10.
    Portilla, J., Simoncelli, E.: A parametric texture model based on joint statistics of complex wavelet coefficients. Int. J. Comput. Vis. 40(1), 49–70 (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    Savarino, F., Hühnerbein, R., Åström, F., Recknagel, J., Schnörr, C.: Numerical integration of Riemannian gradient flows for image labeling. In: Lauze, F., Dong, Y., Dahl, A.B. (eds.) SSVM 2017. LNCS, vol. 10302, pp. 361–372. Springer, Cham (2017). CrossRefGoogle Scholar
  12. 12.
    Werner, T.: A linear programming approach to max-sum problem: a review. IEEE Trans. Patt. Anal. Mach. Intell. 29(7), 1165–1179 (2007)CrossRefGoogle Scholar
  13. 13.
    Xie, J., Lu, Y., Zhu, S.C., Wu, Y.: A theory of generative ConvNet. In: Proceedings of the ICML (2016)Google Scholar
  14. 14.
    Zhu, S., Mumford, D.: Prior learning and Gibbs reaction-diffusion. IEEE Trans. Patt. Anal. Mach. Intell. 19(11), 1236–1250 (1997)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Image and Pattern Analysis GroupHeidelberg UniversityHeidelbergGermany
  2. 2.Biomedical Computer Vision Group, BIOQUANTHeidelberg UniversityHeidelbergGermany

Personalised recommendations